Energy Levels in a Crystalline Solid — Question 8

Question 8

A one-dimensional band accumulates states near its edges

For a long chain use $E=E_0-2t\cos(ka)$, length L and two spins. Define total density of states $D(E)$ such that $D(E)dE$ counts all states in an energy interval, including both k branches.

1. Derive $D(E)$ inside the band.

2. Check the total state count by integrating over the full band.

Question 8 - Solution

A one-dimensional band accumulates states near its edges

(a) Method and calculation

$$\frac{dN}{dk}=\frac{2L}{2\pi}\text{ per branch},\quad\left|\frac{dE}{dk}\right|=a\sqrt{4t^2-(E-E_0)^2}.$$$$\boxed{D(E)=\frac{2L}{\pi a\sqrt{4t^2-(E-E_0)^2}}}.$$

(b) Interpretation and further calculation

$$\int_{E_0-2t}^{E_0+2t}\frac{dE}{\sqrt{4t^2-(E-E_0)^2}}=\pi.$$$$\boxed{\int D(E)dE=2L/a=2N}.$$

Physical check

The edge divergences are integrable and reflect vanishing group velocity. A finite chain has discrete levels; the smooth density is a large-N approximation.